Listed below are the measured radiation emissions​ (in W/kg) corresponding to cell​ phones: A,​ B, C,​ D, E,​ F, G,​ H, I,​ J, and K respectively. The media often present reports about the dangers of cell phone radiation as a cause of cancer. Cell phone radiation must be 1.6​ W/kg or less. Find the a.​ mean, b.​ median, c.​ midrange, and d. mode for the data. Also complete part e.
0.87
0.25
0.77
0.68
0.21
1.07
1.03
1.27
0.92
1.41
0.69
 


Let's calculate the required statistics step by step using the given dataLet's calculate the required statistics step by step using the given 

**Given Data:**
- Radiation emissions in W/kg: 0.87, 0.25, 0.77, 0.68, 0.21, 1.07, 1.03, 1.27, 0.92, 1.41, 0.69

**a. Mean:**

To find the mean, sum all the values and divide by the number of values:

\[
\text{Mean} = \frac{\sum \text{(all values)}}{\text{Number of values}}
\]

\[
\text{Mean} = \frac{0.87 + 0.25 + 0.77 + 0.68 + 0.21 + 1.07 + 1.03 + 1.27 + 0.92 + 1.41 + 0.69}{11}
\]

\[
\text{Mean} = \frac{9.17}{11} \approx 0.834 \, \text{W/kg}
\]

**b. Median:**

To find the median, sort the data in ascending order and find the middle value:

Sorted dataLet's calculate the required statistics step by step using the given 

**Given Data:**
- Radiation emissions in W/kg: 0.87, 0.25, 0.77, 0.68, 0.21, 1.07, 1.03, 1.27, 0.92, 1.41, 0.69

**a. Mean:**

To find the mean, sum all the values and divide by the number of values:

\[
\text{Mean} = \frac{\sum \text{(all values)}}{\text{Number of values}}
\]

\[
\text{Mean} = \frac{0.87 + 0.25 + 0.77 + 0.68 + 0.21 + 1.07 + 1.03 + 1.27 + 0.92 + 1.41 + 0.69}{11}
\]

\[
\text{Mean} = \frac{9.17}{11} \approx 0.834 \, \text{W/kg}
\]

**b. Median:**

To find the median, sort the data in ascending order and find the middle value:

Sorted  0.21, 0.25, 0.68, 0.69, 0.77, 0.87, 0.92, 1.03, 1.07, 1.27, 1.41

Since there are 11 values, the median is the middle value (6th value):

\[
\text{Median} = 0.87 \, \text{W/kg}
\]

**c. Midrange:**

The midrange is the average of the smallest and largest values:

\[
\text{Midrange} = \frac{\text{Smallest value} + \text{Largest value}}{2}
\]

\[
\text{Midrange} = \frac{0.21 + 1.41}{2} = \frac{1.62}{2} = 0.81 \, \text{W/kg}
\]

**d. Mode:**

The mode is the value that appears most frequently in the data set. Since all values are unique, there is no mode for this data set.

\[
\text{Mode} = \text{None (no repeated values)}
\]

**e. Additional Question:**

If the task requires an interpretation or analysis, we could evaluate whether the mean, median, or midrange values suggest that the phones are within safe radiation levels according to the 1.6 W/kg threshold.

**Conclusion:**
- All phones have radiation emissions below the 1.6 W/kg limit, indicating compliance with safety standards.

Would you like a more detailed explanation, or do you have any questions?

**Related Questions:**
1. How would removing the highest radiation emission (1.41 W/kg) affect the mean?
2. What percentage of the phones are below the mean radiation emission?
3. How would the mean change if one phone emitted double its current radiation?
4. Is the median always a better measure than the mean for this type of data?
5. What implications might there be if one phone had a significantly higher radiation level, say 2.5 W/kg?

**Tip:** The median is often preferred over the mean in skewed data because it better represents the central tendency by not being affected by extreme values.

Learn how to calculate the mean, median, midrange, and mode of cell phone radiation emissions using a step-by-step approach. This problem involves descriptive statistics and is suitable for students in Grades 10-12.

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